Schauder Estimates for a Class of Potential Mean Field Games of Controls

An existence result for a class of mean field games of controls is provided. In the considered model, the cost functional to be minimized by each agent involves a price depending at a given time on the controls of all agents and a congestion term. The existence of a classical solution is demonstrated with the Leray-Schauder theorem; the proof relies in particular on a priori bounds for the solution, which are obtained with the help of a potential formulation of the problem.


Introduction
The goal of this work is to prove the existence and uniqueness of a classical solution to the following system of partial differential equations: −∂ t u − σ∆u + H(x, t, ∇u(x, t) + φ(x, t) P (t)) = f (x, t, m(t)) (x, t) ∈ Q, (ii) ∂ t m − σ∆m + div(vm) = 0 (x, t) ∈ Q, (iv) v(x, t) = −H p (x, t, ∇u(x, t) + φ(x, t) P (t)) (x, t) ∈ Q, with (x, t) ∈ Q := T d × [0, T ]. The parameters T > 0, σ > 0 are given and are given data. The set D 1 (T d ) is defined as Here we work with Z d -periodic data and we set the state set as the d-dimensional torus T d , that is a quotient set R d /Z d . The Hamiltonian H is assumed to be such that H(x, t, p) = L * (x, t, −p), for some mapping L, where L * (x, t, −p) denotes the Fenchel transform with respect to p. The mapping L is assumed to be convex in its third variable. The function u, as a solution to the Hamilton-Jacobi-Bellman (HJB) in equation (i)(MFGC) is the value function corresponding to the stochastic optimal control problem: inf α E T 0 L(X t , t, α t ) + φ(X t , t) P (t), α t + f (X t , t, m(t)) dt + g(X T ) , subject to the stochastic dynamics dX t = α t dt + √ 2σ dB t , X 0 = x ∈ T d . The feedback law v given by (iv)(MFGC) is then optimal for this stochastic optimal control problem. Equation (ii)(MFGC) is the Fokker-Planck equation which describes the evolution of m t = L(X t ) when the optimal feedback law is employed. At last, (iii)(MFGC) makes the quantity P (t) endogenous.
An interpretation of the system (MFGC) is as follows. Consider a stock trading market. A typical trader, with an initial level of stock X 0 = x, controls its level of stock (X t ) t∈[0,T ] through the purchasing rate α t with stochastic dynamic dX t = α t dt + √ 2σdB t . The agent aims at minimizing the expected cost (2) where P (t) is the price of the stock at time t. The agent is considered to be infinitesimal and has no impact on P (t), so it assumes the price as given in its optimization problem. On the other hand, in the equilibrium configuration, the price P (t) (t ∈ [0, T ]) becomes endogenous and indeed, is a function of the optimal behaviour of the whole population of agents as formulated in (iii)(MFGC). The expression D(t) := T d φ(x, t)v(x, t)m(x, t) dx can be considered as a weigted net demand formulation and the relation P = Ψ(D) is the result of supply-demand relation which determines the price of the good at the market. Thus, this system captures an equilibrium configuration. Similar models have been proposed in the electrical engineering literature, see for example [2,10,11] and the references therein.
In most mean field game models, the individual players interact through their position only, that is, via the variable m. The problem that we consider belongs to the more general class of problems, called extended mean field games, for which the players interact through the joint probability distribution µ of states and controls. Several existence results have been obtained for such models: in [13] for stationary mean field games, in [15] for deterministic mean field games. In [6,Section 5], a class of problems where µ enters in the drift and the integral cost of the agents is considered. We adopt the terminology mean field games of controls employed by the authors of the latter reference. Let us mention that our existence proof is different from the one of [6], which includes control bounds. In [3, Section 1], a model where the drift of the players depends on µ is analyzed. In [14], a mean field game model is considered where at all time t, the average control (with respect to all players) is prescribed. We finally mention that extended mean field games have been studied with a probabilistic approach in [1,8] and in [7,Section 4.6], and that a class of linear-quadratic extended mean field games has been analyzed in [19].
A difficulty in the study of mean field games of controls is the fact that the control variable, at a given time t, cannot be expressed in an explicit fashion as a function of m(·, t) and u(·, t).
Instead, one has to analyze the well-posedness and the stability of a fixed point equation (see for example [6,Lemma 5.2]). In our model, if we combine (iii) and (iv)(MFGC), we obtain the fixed point equation v = −H p (∇u + Ψ(∫ φvm)) for the control variable v. A central idea of the present article is the following: equation (3) is equivalent to the optimality conditions of a convex optimization problem, when L is convex and Ψ is the gradient of a convex function Φ. This observation allows to show the existence and uniqueness of a solution v (to equation (3)) and to investigate its dependence with respect to ∇u and m in a natural way. More precisely, we prove that this dependence is locally Hölder continuous. The existence of a classical solution of (MFGC) is established with the Leray-Schauder theorem and classical estimates for parabolic equations. A similar approach has been employed in [16] and [17] for the analysis of a mean field game problem proposed by Chan and Sircar in [9]. In this model, each agent exploits an exhaustible resource and fixes its price. The evolution of the capacity of a given producer depends on the price set by the producer, but also on the average price (with respect to all producers).
The application of the Leray-Schauder theorem relies on a priori bounds for fixed points. These bounds are obtained in particular with a potential formulation of the mean field game problem: we prove that all solutions to (MFGC) are also solutions to an optimal control problem of the Fokker-Planck equation. We are not aware of any other publication making use of such a potential formulation for a mean field game of controls, with the exception of [17] for the Chan and Sircar model. Let us mention that besides the derivation of a priori bounds, the potential formulation of the problem can be very helpful for the numerical resolution of the problem and the analysis of learning procedures (which are out of the scope of the present work).
The article is structured as follows. We list in Section 2 the assumptions employed all along. The main result (Theorem 3.1) is stated in Section 3. We provide in Section 4 a first incomplete potential formulation of the problem, incomplete in so far as the term f (m) is not integrated. We also introduce some auxiliary mappings, which allow to express P and v as functions of m and u. We give some regularity properties for these mappings in Section 5. In Section 6 we establish some a priori bounds for solutions to the coupled system. We finally prove our main result in Section 7. In the last section, we give a full potential formulation of the problem, prove the uniqueness of the solution to (MFGC) and prove that (u, P, f (m)) is the solution to an optimal control problem of the HJB equation, under an additional monotonicity condition on f .
2 Assumptions on data and parabolic estimates

Notation and assumptions
Let us introduce the main notation used in the article. Recall that D 1 (T d ) was defined in (1). For all m ∈ D 1 (T d ), for all measurable functions v : T d → R d such that |v(·)| 2 m(·) is integrable, the following inequality holds true, by the Cauchy-Schwarz inequality.
The gradient of the data functions with respect to some variable is denoted with an index, for example, H p denotes the gradient of H with respect to p. The same notation is used for the Hessian matrix. The gradient of u with respect to x is denoted by ∇u. Let us mention that very often, the variables x and t are omitted, to alleviate the calculations. We also denote by φvm the integral T d φvm dx when used as a second argument of Ψ. For a given normed space X, the ball of center 0 and radius R is denoted B(X, R).
Along the article, we use the following Hölder spaces: C α (Q), C 2+α (T d ), and C 2+α,1+α/2 (Q), defined as usual with α ∈ (0, 1). Sobolev spaces are denoted by W k,p , the order of derivation k being possibly non-integral. We fix now a real number p such that p > d + 2.
We will also make use of the following Banach space: Convexity assumptions We collect below the required assumptions on the data. As announced in the introduction, H is related to the convex conjugate of a mapping L : Q × R d → R as follows: The mapping L is assumed to be strongly convex in its third variable, uniformly in x and t, that is, we assume that L is differentiable with respect to v and that there exists C > 0 such that for all (x, t) ∈ Q and for all v 1 and v 2 ∈ R d . This ensures that H takes finite values and that H is continuously differentiable with respect to p, as can be easily checked. Moreover, the supremum in (5) is reached for a unique v, which is then given by We also assume that Ψ has a potential, that is, there exists a mapping Φ : [0, T ] × R k → R, differentiable in its second argument, such that Regularity assumptions We assume that L v is differentiable with respect to x and v and that φ is differentiable with respect to x. All along the article, we make use of the following assumptions.
We finally assume that Let us mention here that the variables C > 0 and α ∈ (0, 1) used all along the article are generic constants. The value of C may increase from an inequality to the next one and the value of the exponent α may decrease.
Some lower bounds for L and for Φ can be easily deduced from the convexity assumptions. By Assumption (A6), L(x, t, 0) and L v (x, t, 0) are bounded. It follows then from the strong convexity assumption (A1) and (A2) that there exists a constant C > 0 such that Without loss of generality, we can assume that Φ(t, 0) = 0, for all t ∈ [0, T ]. Since Φ is convex, we have that Φ(t, z) ≥ Ψ(t, 0), z , for all z ∈ R k . We deduce then from Assumption (A4) that where C is independent of t and z. Some regularity properties for the Hamiltonian can be deduced from the convexity assumption (A1) and the Hölder continuity of L and its derivatives (Assumption (A6)). They are collected in the following lemma.
Lemma 2.1. The Hamiltonian H is differentiable with respect to p and H p is differentiable with respect to x and p. Moreover, for all R > 0, there exists α ∈ (0, 1) such that Proof. For a given (x, t, p) ∈ Q × R d , there exists a unique v := v(x, t, p) maximizing the function v ∈ R d → − p, v − L(x, t, p), which is strongly concave by (A1). It is then easy to deduce from (8) and the boundedness of L(x, t, 0) that there exists a constant C, independent of (x, t, p), such that |v(x, t, p)| ≤ C(|p| + 1).
For all (x, t, p) ∈ Q × R d , Since L v is continuously differentiable with respect to x and v, we obtain with the inverse mapping theorem that v(x, t, p) is continuously differentiable with respect to x and p. Let R > 0 and let (x 1 , t 1 , p 1 ) and (x 2 , t 2 , where C does not depend on x i , t i , and p i (but depends on R). Moreover, we have We deduce from (A1), Young's inequality, and (A6) that there exists C > 0 and α ∈ (0, 1), both independent of x i , t i , and p i (but dependent of R) such that for all ε > 0. Taking ε = 1 2C , we deduce that the mapping (x, t, p) ∈ B R → v(x, t, p) is Hölder continuous. Since L is Hölder continuous on bounded sets, we obtain that the Hamiltonian One can easily check that H p (x, t, p) = −v(x, t, p), which proves that H p is Hölder continuous on B R . Finally, differentiating relation (11) with respect to x and p, we obtain that We deduce then with Assumption (A6) that D x v(x, t, p) and D p v(x, t, p) (and thus H px and H pp ) are Hölder continuous on B R , as was to be proved.
An example of coupling term We finish this subsection with an example of a mapping f satisfying the regularity assumptions (A5) and (A7). Let ϕ ∈ L ∞ (R d ) be a given Lipschitz continuous mapping, with modulus C 1 . Let us set C 2 = ϕ L ∞ (R d ) . Let K : Q × [−C 2 , C 2 ] → R be a measurable mapping satisfying the following assumptions: 2. There exist a mapping C 3 ∈ L 1 (T d ) and α ∈ (0, 1) such that for a.e. x ∈ T d , for all t 1 and t 2 ∈ [0, T ] and for all w 1 and Let us setφ(x) := ϕ(−x). We identify m ∈ L ∞ (T d ), with its extension by 0 over R d so that the convolution product below is well-defined: We keep in mind that m * ϕ is a function over T d . Then In a similar way we can define and we have that The following lemma is inspired from [4, Example 1.1].

Main result and general approach
Theorem 3.1. There exists α ∈ (0, 1) such that (MFGC) has a classical solution (m, u, P, v), with The result is obtained with the Leray-Schauder theorem, recalled below.
Theorem 3.2 (Leray-Schauder, [12], Theorem 11.6). Let X be a Banach space and let T : X × [0, 1] → X be a continuous and compact mapping. Assume that T (x, 0) = 0 for all x ∈ X and assume there exists C > 0 such that The application of the Leray-Schauder theorem and the construction of T will be detailed in Section 7. Let us mention that the set of fixed points of T (·, τ ), for τ ∈ [0, 1], will coincide with the set of solutions of the following parametrization of (MFGC): (MFGC τ ) Of course, (MFGC τ ) corresponds to (MFGC) for τ = 1. Let us introduce the space X and X , used for the formulation of the fixed-point equation: The HJB equation (i) and the Fokker-Planck equation (ii) are classically understood in the viscosity and weak sense, respectively. However, due to the choice of the solution spaces, we may interpret these equations as equalities in L p (Q). A first and important step of our analysis is the construction of auxiliary mappings allowing to express v and P as functions of m and u. These mappings cannot be obtained in a straightforward way, since in (iii), P depends on v and in (iv), v depends on P .
∇m(x, t), so that integrating (by parts) over Q t := T d × (0, t), since v is essentially bounded, we get that so that after cancellation of the contribution of ∇µ, we obtain, applying Gronwall's lemma to a(t) : Integrating by parts the double integral we see that it is equal to 0, and we conclude by noting that

Potential formulation
In this section, we first establish a potential formulation of the mean field game problem (MFGC τ ), that is to say, we prove that for (u τ , m τ , v τ , P τ ) ∈ X satisfying (MFGC τ ), (m τ , v τ ) is a solution to an optimal control problem. We prove then that for all t, v τ (·, t) is the unique solution of some optimization problem, which will enable us to construct the announced auxiliary mappings. Let us introduce the cost functional B : We have the following result.
is the solution to the following optimization problem: Remark 4.1. Let us emphasize that the above optimal control problem is only an incomplete potential formulation, since the termf τ still depends on m τ .
. Therefore, by (5) and (6), we have that for all (x, t) ∈ Q. Moreover, by Lemma 3.1, m ≥ 0 and m τ ≥ 0. Therefore, Using (i)(MFGC τ ), we obtain After integration with respect to x, we obtain that for all t, We obtain with the convexity of Φ and (iii)(MFGC τ ) that Using the previous calculations to bound B(m, v;f τ ) − B(m τ , v τ ;f τ ) from below, we observe that the term P τ , φ(m − m τ v τ ) cancels out and obtain that Integrating by parts and using (ii)(MFGC τ ), we finally obtain that as was to be proved. We do not detail the proof for the case τ = 0, which is actually simpler. Indeed, for τ = 0, the solution to the Fokker-Planck equation is independent of v and thus m = m τ in the above calculations.
We have proved that the pair (m τ , v τ ) is the solution to an optimal control problem. Therefore, for all t, v τ (·, t) minimizes the Hamiltonian associated with problem (18). Let us introduce some notation, in order to exploit this property.
Combining inequalities (19) and (20) The following lemma will enable us to express P τ (t) and v τ (·, t) as functions of m τ (·, t) and u τ (·, t). The key idea is, roughly speaking, to prove the existence and uniqueness of a minimizer to J(·; t, m, w). The where the constant C is independent of t, m, and w (but depends on R).
Proof. If the pair (v, P ) satifies (21), then One can easily check that for proving the existence and uniqueness of a pair (v, P ) satisfying (21), it is sufficient to prove the existence and uniqueness of v ∈ L ∞ (T d , R d ) satisfying (23). For future reference, let us mention that by (6), relation (23) is equivalent to Step 1 : existence and uniqueness of a minimizer of J( Since the sum of a l.s.c. convex function and of a l.s.c. strongly convex function is l.s.c. and strongly convex, so is J(·; t, m, w). Since it is continuous, it possesses a unique minimizerv in L 2 with C independent of t, m, and w, but depending on R, as all constants C used in the proof.
Step 2: existence of v(t, m, w) and a priori bound.
Using then the equivalence of (23) and (24), we obtain that

Consider now the measurable function
The two functions v andv may not be equal for a.e. x if m(x) = 0 on a subset of T d of non-zero measure. Still they are equal in L 2 m (T d , R d ), which ensures in particular that φ(x , t)v(x )m(x ) dx = φ(x , t)v(x )m(x ) dx and finally that v satisfies (23) and lies in L ∞ (T d , R d ), as a consequence of the continuity of H p (proved in Lemma 2.1). We also have that v L 2 (25). Using the Cauchy-Schwarz inequality and Assumption (A6), we obtain that | φvm| ≤ C. We obtain then with Assumption (A4) that for P = Ψ( φvm), we have |P | ≤ C. Using Assumption (A6) and the continuity of H p , we finally obtain that v L ∞ (T d ,R d ) ≤ C. Thus the bound (22) is satisfied.

Regularity results for the auxiliary mappings
We provide in this section some regularity results for the mappings v and P. We begin by proving that P(·, ·, ·) is locally Hölder continuous. For this purpose, we perform a stability analysis of the optimality condition (24).
Then, there exist a constant C > 0 and an exponent α ∈ (0, 1), both independent of t 1 , t 2 , w 1 , w 2 , m 1 , and m 2 but depending on R, such that Proof. Note that all constants C > 0 and all exponents α ∈ (0, 1) involved below are independent of t 1 , t 2 , w 1 , w 2 , m 1 , and m 2 . They are also independent of x ∈ T d and ε > 0. For i = 1, 2, we set By the optimality condition (24), we have that Consider the difference of (28) for i = 2 with (28) for i = 1. Integrating with respect to x the scalar product of the obtained difference with v 2 m 2 − v 1 m 1 , we obtain that We look for a lower estimate of these three terms. Let us mention that the term v 2 m 2 − v 1 m 1 , appearing in the three terms, will be estimated only at the end. Estimation from below of (a 1 ). We have (a 1 ) = (a 11 ) + (a 12 ), where By monotonicity of Ψ, we have that Let us consider (a 11 ). We set and further with Young's inequality that Estimation from below of (a 2 ). We have (a 2 ) = (a 21 ) + (a 22 ) + (a 23 ), where As a consequence of (27), assumption (A6), and Young's inequality, we have that By (27) and assumption (A6), Finally, we have by assumption (A1) and since m 1 ≥ 0. Estimation from below of (a 3 ). Using (30) and Young's inequality, we obtain that Let us estimate v 2 m 2 − v 1 m 1 L 1 (T d ;R d ) . Using the Cauchy-Schwarz inequality, we obtain that Injecting this inequality in (29) and taking ε = 1 3C , we obtain that Let us prove (26). We have Therefore, using assumption (A6) and (31), we obtain that Inequality (26) follows, using assumption (A6).
Lemma 5.2. For all R > 0, the mapping and the mapping are both Hölder continuous, that is, there exist α ∈ (0, 1) and C > 0 such that for all m 1 and m 2 ∈ L ∞ (0, T ; D 1 (T d )), for all w 1 and w 2 ∈ B(L ∞ (Q, R d ), R), and for all u 1 and u 2 in B(W 2,1,p (Q), R).
Proof. The Hölder continuity of the first mapping is a direct consequence of Lemma 5.1. As a consequence, the mapping is Hölder continuous. Using then the relations and the Hölder continuity of H p , H px , and H pp on bounded sets (Lemma 2.1), we obtain that the second mapping is Hölder continuous.
Proof. We recall that by Lemma A.2, ∇u C α (Q,R d ) ≤ C u W 2,1,p (Q) . We obtain then the bound on P(m, ∇u) C α (0,T ;R k ) with Lemma 5.1.
Step 1: P τ L 2 (0,T ;R k ) ≤ C. Let us take v 0 = 0 and let m 0 be the solution to the heat equation by the Cauchy-Schwarz inequality and Young's inequality. The constant C is also independent of ε. Using then the lower bounds (8) and (9) and assumptions (A5) and (A8), we obtain that Taking ε = 1/(2C 2 ), we deduce that Q |v τ | 2 m τ dx dt ≤ C. Using then assumption (A4), the boundedness of φ, the Cauchy-Schwarz inequality and the estimate obtained previously, we deduce that as was to be proved.
Step 2: u τ L ∞ (Q) ≤ C, ∇u τ L ∞ (Q,R d ) ≤ C. The argument is classical. We have that u τ is the unique solution to the HJB equation (i)(MFGC τ ). It is therefore the value function associated with the following stochastic optimal control problem: where and (X s ) s∈[t,T ] is the solution to the stochastic dynamic dX s = τ α s ds + √ 2σdB s , X t = x. Here, L 2 F (t, T ; R d ) denotes the set of stochastic processes on (t, T ), with values in R d , adapted to the filtration F generated by the Brownian motion (B s ) s∈[0,T ] , and such that E T t |α(s)| 2 ds < ∞. Then, the boundedness of u τ from above can be immediately obtained by choosing α = 0 in (36) and using the boundedness of g. We can as well bound u τ from below since for all (x, s) ∈ Q and for all α ∈ R d , we have for some constant C independent of (x, s), α, and P τ (s). We already know from the previous step that P τ L 2 (0,T ;R k ) ≤ C. So we can conclude that u τ is also bounded from below, and thus u τ L ∞ (Q) ≤ C. We also deduce from the above inequality that for all α ∈ L 2 F (t, T ; R d ), Let us bound ∇u τ . Choose ε ∈ (0, 1). For arbitrary (x, t), take an ε-optimal stochastic optimal controlα for (36). We can deduce from the boundedness of the map u τ and inequality (38) that where C is independent of (τ, x, t) and ε. Let y ∈ T d . Set then obviously dY s =α s ds + √ 2σdB s , Y t = y. We have where (a), (b), (c), (d) are given by as a consequence of assumption (A3) and (39). Then, using assumption (A6), (35), and (39), we obtain By assumption (A8), |(c)| ≤ E |g(Y T ) − g(X T )| ≤ C|y − x|. Finally, sincef τ is a Lipschitz function (by assumption (A7)), Letting ε → 0, we obtain that u τ (y, t) − u τ (x, t) ≤ C|y − x|. Exchanging x and y, we obtain that u τ is Lipschitz continuous with modulus C and finally that ∇u τ L ∞ (Q,R d ) ≤ C.
Step 6: m τ C α (Q) ≤ C. The Fokker-Planck equation can be written in the form of a parabolic equation with coefficients in L p : Combining Theorem A.2 and lemma A.2, we get that m τ C α (Q) ≤ C.

Application of the Leray-Schauder theorem
Proof of Theorem 3.1.
Step 1: construction of T . Let us define the mapping T : X × [0, 1] → X which is used for the application of the Leray-Schauder theorem. A difficulty is that the auxiliary mappings P and v are only defined for m ∈ L ∞ (0, T ; D 1 (T d )). Therefore we need a kind of projection operator on this set. Note that T d 1 dx = 1. We consider the mapping where m + (x, t) = max(0, m(x, t)). The well-posedness of the mapping can be easily checked, as well as the following properties: • For all m ∈ L ∞ (0, T ; D 1 (T d )), ρ(m) = m.
For proving the first property, we suggest to consider separately the two cases: m + (y, t) dy < 1 and m + (y, t) dy ≥ 1. For a given (u, m, τ ) ∈ X × [0, 1], the pair (ũ,m) = T (u, m, τ ) is defined as follows:ũ is the solution to andm is the solution to Let us observe that T (u, m, 0) is not necessarily null, but is independent of (u, m). Thus the Leray-Schauder theorem is still valid, as can be seen with a translation argument that we do not detail.
Step 3: continuity of T . Using the continuity of ρ, Lemma 5.2, the Hölder continuity of H, and assumption (A7), we obtain that the mappings are continuous. By Theorem A.4, the solution to a parabolic equation of the form (53), with b and c null (in W 2,1,p (Q)) is a continuous mapping of the right-hand side (in L p (Q)). Thus, u ∈ W 2,1,p (Q) andm ∈ W 2,1,p (Q) depend in a continuous way on H(∇u + φ P(ρ(m), ∇u)) and div(v(ρ(m), ∇u)m), which finally proves that T is the composition of continuous mappings and thus is continuous itself.
Step 5: conclusion. The existence of a fixed point (u, m) to T (·, ·, 1) follows. By the maximum principle (Lemma 3.1), we have that m(t) ∈ D 1 (T d ) for all t and thus ρ(m(t)) = m(t) for all t. With the same arguments as those used before, we obtain that (u, m, P(m, ∇u), v(m, ∇u)) is a solution to (MFGC τ ) with τ = 1 and finally that (16) holds, by Proposition 6.1.

Uniqueness and duality
In this section we prove the uniqueness of the solution (u, m, P, v) to (MFGC). We also prove that (P, v) is the solution to a dual problem to (18). Both results are obtained under the following additional monotonicity assumption of f : There exists a measurable mapping F (t, m) for all m 1 and m 2 ∈ D 1 (T d ) and for a.e. t. Thus, F (t, ·) is a supremum of the exact affine minorants appearing in the above right-hand side, and is therefore a convex function of m.
Remark 8.1. 1. It follows from (42) that f is monotone: for all m 1 and m 2 ∈ D 1 (T d ) and for a.e. t. Conversely, (42) holds true if (43) is satisfied and if F is a primitive of f (., t, .) in the sense that We refer to [5, Proposition 1.2] for a further characterization of functions f deriving from a potential.

2.
Consider the mapping f K proposed in Lemma 2.2. Assume that for all (x, t) ∈ Q, K(x, t, ·) is non-decreasing and consider the function K, defined by K(x, t, w) : . Then inequality (42) holds true with F K defined by Indeed, since K is convex in its third argument, we have as was to be proved.
Without loss of generality, we can assume that F (t, m 0 ) = 0 for a.e. t ∈ (0, T ). It can then be easily deduced from assumption (A5) and (42) that there exists a constant C such that Let us consider the potential B : Proposition 8.1. There exists a unique solution (u, m, P, v) ∈ X to (MFGC). Moreover, the pair (m, v) is the unique solution to the following optimal control problem min Proof. Let (u, m, P, v) ∈ X be a solution to (MFGC). Let us prove that (m, v) is a solution to (46). Let (m,v) be a feasible pair. Denotingf (x, t) = f (x, t, m(t)), we have The two terms in the right-hand side are both nonnegative, as a consequence of Lemma 4.1 and assumption (42), respectively.
To conclude the proof of the proposition, it suffices to prove that (46) has a unique solution. Let us prove first a classical property: There exists a constant C > 0 such that for all (x, t) ∈ Q, for all p ∈ R d and for all v ∈ R d , Let us setv = −H p (x, t, p). For a fixed triple (x, t, p), Inequality (47) follows. Let (u 1 , m 1 , P 1 , v 1 ) and (u 2 , m 2 , P 2 , v 2 ) be two solutions to (MFGC) in X . We obtain with inequality (47) that Proceeding then exactly like in the proof of Lemma 4.1, we arrive at the following inequality: We also have that Let us set m = m 2 − m 1 . Using relation (48), we obtain that m is the solution to the following parabolic equation: Therefore m = 0 and m 2 = m 1 . We already know that v 2 m 2 = v 1 m 2 , we deduce then that v 2 m 2 = v 1 m 1 . We obtain further with (iii) that P 1 = P 2 , then with (i) that u 1 = u 2 and finally with (iv) that v 1 = v 2 , which concludes the proof.
We finish this section with a duality result. For γ ∈ L ∞ (T d ), we recall that the convex conjugate of F (t, ·) is defined by It directly follows from the above definition that |F * (t, γ)| ≤ γ L ∞ (T d ) +C, where C is the constant obtained in (44) and thus for γ ∈ L ∞ (Q), T 0 F * (t, γ(·, t)) dt is well-defined. Consider the dual criterion D : (u, P, γ) ∈ W 2,1, The function Φ * is the convex conjugate of Φ with respect to its second argument. Since Φ(t, 0) = 0, we have that Φ * (t, ·) ≥ 0 and thus the first integral is well-defined in R ∪ {∞}.
Lemma 8.1. Let (ū,m,v,P ) be the solution to (MFGC). Letf be defined byf (x, t) = f (x, t,m(t)). Then, (ū,P ,f ) is the solution to the following problem: D(u, P, γ), subject to: Moreover, for all solutions (u, P, γ) to the dual problem, P =P . If in addition, γ =f and the above inequalities hold as equalities, then u =ū.
We also have that where Integrating by parts (in time), we obtain that Integrating by parts (in space), we further obtain that where Combining (50), (51) and (52) together, we finally obtain that Therefore, for all optimal solutions (u, P, γ), P =P . If moreover γ =f and the inequality constraints in (49) hold as equalities, then (since the HJB equation has a unique solution) u =ū, which concludes the proof.

Conclusion
The existence and uniqueness of a classical solution to a mean field game of controls have been demonstrated. A particularly important aspect of the analysis is the fact that the equations (iii) and (iv)(MFGC), encoding the coupling of the agents through the controls, are equivalent to the optimality system of a convex problem. This observation has eventually enabled us to eliminate the variables v and P from the coupled system. The analysis done in this article can be extended in different ways. A more complex interaction between the agents could be considered. For example, it would be possible to replace equations (iii) and (iv) by the following ones: assuming that ϕ is convex with respect to v and Ψ ≥ 0. For a fixed t ∈ [0, T ], this system is equivalent to the optimality system associated with the following convex problem: Another possibility of extension of our analysis would be to add convex constraints on the control variable.
Future research will aim at exploiting the potential structure of the problem, which can be used to solve it numerically and to prove the convergence of learning procedures, as was done in [5].

A A priori bounds for parabolic equations
In this appendix we provide estimates for the following parabolic equation: for different assumptions on b, c, h, and u 0 . The technique is based on the following idea. By standard parabolic estimates detailed below, (53) has a unique solution u in L 2 (0, T ; H 1 (T d )), that we may identify with a periodic function over R d . Let ϕ : R d → R be of class C ∞ , with value 1 in a neighbourhood of the closure of T d , and with compact support in Ω := B(0, 2). Set Q := Ω×(0, T ).
Observe that the solution v of (54) is equal to 0 in a vicinity of (∂Ω) × (0, T ), and hence, satisfies the homogeneous Neumann condition; this allows us to apply some results of [18].
Theorem A.2. Let p > d+2. For all R > 0, there exists C > 0 such that for all u 0 ∈ W 2−2/p,p (T d ), for all b ∈ L p (Q, R d ), for all c ∈ L p (Q), for all h ∈ L p (Q), satisfying equation (53) has a unique solution u in W 2,1,p (Q) which moreover satisfies u W 2,1,p (Q) ≤ C.
We construct now q k+1 in such a way that h [u] ∈ L q k+1 (Q ). Since v ∈ W 2,1,q k (Q ), we have that u ∈ W 2,1,q k (Q) and thus by Lemma A.1, b, ∇u ∈ L r (Q ) with If q k < 1 + d/2, then cu ∈ L r (Q ) with Note that r > r . If q k ≥ 1 + d/2, then u ∈ L ∞ (Q ) and thus cu ∈ L p (Q ). We set now q k+1 = min(r , p). We observe that in both cases, cu ∈ L q k+1 (Q ). One can verify that the other terms of h [u] also lie in L q k+1 (Q ). Therefore, by Theorem A.1, v ∈ W 2,1,q k+1 (Q ). If q k+1 ≥ d + 2, we stop the construction of the sequence and set K = k+1. It remains to prove that the construction of the sequence stops after finitely many iterations. If that was not the case, we would have that q k+1 = r , with r defined in (62), for all k ∈ N, implying that which is a contradiction. Now we know that v ∈ W 2,1,q K (Q ), with q K ≥ d + 2. This implies that u ∈ L ∞ (Q ) and ∇u ∈ L ∞ (Q , R d ) (by Lemma A.1) and thus that h [u] ∈ L p (Q ). Finally, v ∈ W 2,1,p (Q ) (by Theorem A.1) and u ∈ W 2,1,p (Q), since u and v coincide on Q.
Observing that q 0 ,...,q K only depend on p and d, the reader can check that v (and thus u) can be bounded in W 2,1,p (Q ) by a constant depending on R only. Theorem A.4. Let p > d + 2. There exists a constant C > 0 such that for all u 0 ∈ W 2−2/p,p (T d ) and for all h ∈ L p (Q), the unique solution u to (53) (with b = 0 and c = 0) satisfies the following estimate: Proof. Consider the mapping u ∈ W 2,1,p (Q) → (u(·, 0), ∂ t u − σ∆u − h) ∈ W 2−2/p,p (Ω), L p (Q)).
By Theorem A.3, it is continuous and by Theorem A.4, it is bijective. As a consequence of the open mapping theorem, its inverse is also continuous. The result follows.
Proof. By Theorem IV.5.1, page 320 in [18], the result holds for the homogeneous Neumann condition. By lemma A.2, h[u] is Hölder continuous, so that the result holds for v. Since u and v coincide on T d , the conclusion follows.